[Paper Review] Non-Arbitrage up to Random Horizon and after Honest Times for Semimartingale Models

Tahir Choulli, Anna Aksamit+2 more|arXiv (Cornell University)|Oct 4, 2013

Stochastic processes and financial applications25 references6 citations

TL;DR

This paper investigates how stopping a semimartingale financial model at a random time affects its No-Unbounded-Profit-with-Bounded-Risk (NUPBR) property, a key non-arbitrage condition. Using progressive enlargement of filtration and stochastic calculus, it characterizes pairs of market models and random times that preserve NUPBR, and constructs explicit deflators for local martingales stopped at such times.

ABSTRACT

This paper addresses the question of how an arbitrage-free semimartingale model is affected when stopped at a random horizon. We focus on No-Unbounded-Profit-with-Bounded-Risk (called NUPBR hereafter) concept, which is also known in the literature as the first kind of non-arbitrage. For this non-arbitrage notion, we obtain two principal results. The first result lies in describing the pairs of market model and random time for which the resulting stopped model fulfills NUPBR condition. The second main result characterises the random time models that preserve the NUPBR property after stopping for any market model. These results are elaborated in a very general market model, and we also pay attention to some particular and practical models. The analysis that drives these results is based on new stochastic developments in semimartingale theory with progressive enlargement. Furthermore, we construct explicit martingale densities (deflators) for some classes of local martingales when stopped at random time.

Motivation & Objective

• To determine under what conditions a semimartingale model remains NUPBR when stopped at a random time.
• To identify random time models that preserve the NUPBR property regardless of the underlying market model.
• To develop a general theoretical framework using progressive enlargement of filtration for analyzing stopped semimartingale models.
• To construct explicit martingale densities (deflators) for local martingales stopped at random times in practical models.

Proposed method

• Employing progressive enlargement of filtration to model the random time's information arrival in the market filtration.
• Applying stochastic calculus for semimartingales to analyze the local martingale properties of wealth processes under stopping.
• Deriving conditions under which the strict local martingale property is preserved or lost after stopping at a random time.
• Constructing explicit deflators (martingale densities) for local martingales stopped at random times using the theory of honest times.
• Utilizing the concept of honest times to characterize the behavior of deflators and non-arbitrage conditions in stopped models.
• Analyzing specific practical models to illustrate the general results and validate the theoretical framework.

Experimental results

Research questions

• RQ1 Which combinations of semimartingale market models and random times preserve the NUPBR condition after stopping?
• RQ2 What characterizes random times that preserve NUPBR for any underlying semimartingale market model?
• RQ3 How can explicit deflators be constructed for local martingales stopped at random times in semimartingale models?
• RQ4 What role does progressive enlargement of filtration play in preserving or disrupting non-arbitrage properties under random stopping?
• RQ5 How do honest times influence the martingale and local martingale properties of wealth processes in stopped models?

Key findings

• The paper identifies necessary and sufficient conditions on the random time and the underlying semimartingale model for the stopped process to satisfy the NUPBR condition.
• It characterizes random times—specifically those that are honest times—for which the NUPBR property is preserved universally across all semimartingale models.
• Explicit martingale densities (deflators) are constructed for local martingales stopped at random times, particularly in the context of honest times.
• The results show that the progressive enlargement of filtration framework is essential for analyzing the non-arbitrage properties of stopped models.
• The framework applies to both general and specific practical models, demonstrating robustness and applicability in real-world financial modeling.
• The analysis reveals that the preservation of NUPBR after stopping depends critically on the interplay between the information structure of the random time and the semimartingale dynamics.